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and supplied the cases that are not taken notice of by him.

. Every body knows how much the discoveries of Baron Napier contributed to the perfection of trigonometry. By the logarithms, the calculations of plane and spherical trigonometry are performed with the greatest ease ; and by the rules he invented, the numerous cases of spherical trigonometry are easily retained in the memory. Since his time, the principal writers of trigonometry have had chiefy in view how to construct tables of logarithms in the readiest manner, and to adapt them to trigonometry.

The modern discoveries in mathematics, viz. algebra and fluxions, have likewise caused alterations in the method of constructing trigonometrical tables : for, by means of these, the tables can be construct

ed in a much shorter time. It is difficult, - however, to make such as are only learning trigonometry understand how trigono. metrical tables are constructed by algebra and Auxions; and as plane trigonometry is capable of being sufficiently understood, and commonly taught immediately after learning the first fix books of Euclid, it is surely

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proper that a method of constructing tables should be taught at the same time. Whatever method therefore the learned may take in order to construct tables of logarithms, and of fines and tangents, in the most expeditious manner, it is necessary that a method deduced from the first fix books of Euclid, and common arithmetic, be explained to the learner who has advanced no farther : and as the logarithmic fines, tangents, &c. are the logarithms of the natural fines, tangents, &c. a learner can have but a very imperfect understanding of logarithmic fines and tangents before he is acquainted with the natural.

As I believe some trigonometrical tables we have already to be very fufficient, I thought it only necessary to describe methods by which they may be constructed, without enumerating many, or examining which is preferable. What I have faid of the hyperbolical logarithms, is in order to give the learner a view of the origin and progress of logarithms; and I have distinguished it from the other part of the section; because, in order to understand it, the reader must be acquainted with the nigher geometry.

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The scales and the sector are very useful instruments for folving the cases of trigonometry, when no great accuracy is neceffary; I have therefore described how they are constructed and used in trigonometry.

The sections that treat of perspective, and the projection of the sphere, were intended to make spherical trigonometry more easily understood. I have observed, that beginners have commonly great difficulty to under{tand what is the meaning or use of a spherical triangle when they see the representation of it upon a plane, without being taught how to project the different circles of a sphere upon a plane : and because

projection of the sphere is but a particular case of perspective, I found it of great advantage to make the learner first acquainted with perspective, next with the projection of the sphere, and lastly with spherical triangles. Perspective is curious and useful for its own sake; and therefore I have treated it at greater length than is barely necessary for understanding the projection of the sphere.

The reader will fee, that I had no intention by this treatise to supersede the use of any standard book that is taught as a part

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a regular course of geometry ; but rather fuppofed, that the learner is acquainted with several of them before he can benefit any thing by this. In the first part (except what is said of the hyperbolical logarithms) I have supposed the learner acquainted only with the first six books of Euclid, common arithmetic, and the extraction of the square root. I take it for granted, that every

teacher of Euclid explains the principles upon which the rules for extracting the square root are founded when he is teaching the second book of Euclid, and that he explains what is called the rule of three when he is teaching the fifth and fixth books. In the second part, the learner is supposed to understand the eleventh and twelfth books of Euclid ; especially the first twenty propofitions of the eleventh, and Theodosius

upon the sphere, at least the first book of it. In the second part too, I have made some references to fome of the conic-section figures ; particularly to the ellipsis. Because spheric geometry is, in the universities of Scotland at least, taught in the same class with conic sections, I did not think it necessary fo much as to thew how an ellipsis

is described *

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because two distinct parts of a course of geometry should not in the least interfere with one another.

In the sections that treat of the construc, tion of natural fines, tangents, and secants, I have nearly followed the methods of Regiomontanus and Baron Wolfius. But in order to find the sine of 36°, I was under a necessity of inserting four propositions ; two of them from the thirteenth book of Euclid, and two from Ptolemy. Wolfius finds the fine of 36° by means of his own geometry. What I have said of the com, mon logarithms is also according to him.

For what I have said of the hyperbolical logarithms, I was much indebted to the valuable work of the very learned and judicious M. Montucla, intitled, Histoire des Mathématiques, printed at Paris in 1758, in two volumes quarto. He treats that subject

* In a Synopsis of practical mathematics, lately published at Edinburgh, I observed a problem proposed, To describe an ellipfis.” The solution that is given is, to join two equal arcs of equal lesser circles to two equal arcs of two equal greater circles: the figure made up of these four arcs is said to be an ellipsis. It is unneceffary to show, that such a figure cannot be an ellipfis, and that no part of an ellipsis is circular, however great an affinity there is between the two curves.

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